Finite Group Databasebeta

← #2217×D73 #33240410

Subgroups — #2217×D73 #33240410

Order 18688.

754 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD32×D73C211yesmaximal, characteristic, complemented
2C22×C16×D73C211yesmaximal, characteristic, complemented
3#467×D73C211yesmaximal, characteristic, complemented
4#466×D73C211yesmaximal, characteristic
5#22612030C211yesmaximal, characteristic, complemented
6#22612029C211yesmaximal, characteristic, complemented
7#22612028C211yesmaximal, characteristic, complemented
8#22612027C211yesmaximal, characteristic, complemented
9#22612026C211yesmaximal, characteristic, complemented
10#22612025C211yesmaximal, characteristic, complemented
11#22612024C211yesmaximal, characteristic, complemented
12#367×D73C211yesmaximal, characteristic, complemented
13#358×D73C211yesmaximal, characteristic, complemented
14#357×D73C211yesmaximal, characteristic, complemented
15C73×#2217C211yesmaximal, characteristic, complemented
16C2×Q16×D73C2211yescharacteristic
17C2×D8×D73C2211yescharacteristic
18C22×C8×D73C2211yescharacteristic
19C22⋊Q8×D73—21no—
20#121×D73—21no—
21C2×#14556536C2211yescharacteristic, complemented
22C2×#14556535C2211yescharacteristic, complemented
23C2×#14556534C2211yescharacteristic, complemented
24SD32×D73—24no—
25C22×#11924003C2211yescharacteristic, complemented
26C2×C16×D73—24no—
27C2×C16×D73C2211yescharacteristic
28C2×C16×D73C2211yescharacteristic
29C2×C16×D73C2211yescharacteristic, complemented
30C2×C16×D73C2211yescharacteristic, complemented
31#17988829C2211yescharacteristic, complemented
32#17988828C2211yescharacteristic, complemented
33#17988827—21no—
34#17988826—21no—
35#17988825C2211yescharacteristic, complemented
36#17988824C2211yescharacteristic
37#17988823C2211yescharacteristic
38#17988822—21no—
39#17988821—21no—
40#17988820C2211yescharacteristic
41#107×D73C2211yescharacteristic
42#107×D73C2211yescharacteristic
43#103×D73—21no—
44#102×D73—21no—
45#17988415C2211yescharacteristic, complemented
46#17988414C2211yescharacteristic, complemented
47#17988414C2211yescharacteristic, complemented
48#17988392C2211yescharacteristic, complemented
49#17988391C2211yescharacteristic, complemented
50#17988390C2211yescharacteristic, complemented
51#17988389C2211yescharacteristic, complemented
52#17988388C2211yescharacteristic, complemented
53#17988387C2211yescharacteristic, complemented
54C2×SD32×C73C2211yescharacteristic, complemented
55C22×C16×C73C2211yescharacteristic, complemented
56#467×C73C2211yescharacteristic, complemented
57#466×C73C2211yescharacteristic
58#367×C73C2211yescharacteristic, complemented
59#358×C73C2211yescharacteristic, complemented
60#357×C73C2211yescharacteristic, complemented
61C2×Q8×D73—21no—
62C2×D4×D73—21no—
63C2×D4×D73—41no—
64C22×C4×D73D411yescharacteristic
65C2×Dic292C2311yescharacteristic
66C2×#11924018—21no—
67Q16×D73—22no—
68Q16×D73—22no—
69C2×D584C2311yescharacteristic
70C2×#11924013—21no—
71D8×D73—22no—
72D8×D73—22no—
73C22×#10596831D411yescharacteristic
74C2×C8×D73—22no—
75C2×C8×D73—22no—
76C2×C8×D73D411yescharacteristic
77C2×C8×D73D411yescharacteristic
78C2×C8×D73C2311yescharacteristic
79C2×C8×D73C2311yescharacteristic
80#14556571—21no—
81#14556570—21no—
82#14556569—41no—
83#14556568—41no—
84#14556567—21no—
85#14556566—21no—
86#14556565—21no—
87#14556564—41no—
88#14556563—41no—
89#14556562—21no—
90C4⋊C4×D73—21no—
91C4⋊C4×D73—21no—
92C4⋊C4×D73—41no—
93C22⋊C4×D73—41no—
94C22⋊C4×D73—41no—
95#14556536—22no—
96#14556535—22no—
97#14556534—22no—
98C2×#11924003—22no—
99C2×#11924003C2311yescharacteristic
100C2×#11924003C2311yescharacteristic
101C16×D73—22no—
102C16×D73—22no—
103C16×D73—22no—
104C16×D73D414yescomplemented
105#14556525C2311yescharacteristic
106#14556525C2311yescharacteristic
107#14556524—21no—
108#14556524—21no—
109#14556516—21no—
110#14556515—21no—
111#14556514—21no—
112#14556513—21no—
113#14556512—21no—
114#14556511—21no—
115C2×Q16×C73C2311yescharacteristic
116C2×D8×C73C2311yescharacteristic
117C22×C8×C73C2311yescharacteristic
118C22⋊Q8×C73—21no—
119#121×C73—21no—
120SD32×C73—22no—
121C2×C16×C73—22no—
122C2×C16×C73C2311yescharacteristic
123C2×C16×C73C2311yescharacteristic
124#107×C73C2311yescharacteristic
125#107×C73C2311yescharacteristic
126#103×C73—21no—
127#102×C73—21no—
128C23×D73—41no—
129C23×D73D811yescharacteristic
130C2×Dic146—21no—
131C2×Dic146—41no—
132Q8×D73—42no—
133Q8×D73—42no—
134C2×D292—21no—
135C2×D292—41no—
136C2×C22⋊D73—41no—
137C2×C22⋊D73—41no—
138C2×C22⋊D73—41no—
139D4×D73—42no—
140D4×D73—42no—
141D4×D73—84no—
142C22×Dic73D811yescharacteristic
143C2×C4×D73—22no—
144C2×C4×D73—22no—
145C2×C4×D73—41no—
146C2×C4×D73—41no—
147C2×C4×D73—41no—
148C2×C4×D73D811yescharacteristic
149C2×C4×D73D811yescharacteristic
150C2×C4×D73C2×D411yescharacteristic
151C2×C4×D73C2×D411yescharacteristic
152Dic292—21no—
153Dic292—21no—
154#11924018—42no—
155D584—21no—
156D584—21no—
157#11924013—42no—
158C2×#10596831—22no—
159C2×#10596831C2×D411yescharacteristic
160C2×#10596831C2×D411yescharacteristic
161C8×D73—22no—
162C8×D73—21no—
163C8×D73—21no—
164C8×D73—22no—
165C8×D73C2×D412yes—
166C8×D73#4112yes—
167C4⋊Dic73—21no—
168C4⋊Dic73—21no—
169C4⋊Dic73—41no—
170#11924007—41no—
171#11924007—41no—
172#11924007—41no—
173#11924007—41no—
174C22⋊Dic73—41no—
175C22⋊Dic73—41no—
176#11924005—41no—
177#11924005—41no—
178#11924005—41no—
179#11924005—41no—
180#11924003—21no—
181#11924003C2×D412yescomplemented
182C2×Q8×C73—21no—
183C2×D4×C73—21no—
184C2×D4×C73—41no—
185C22×C4×C73C2×D411yescharacteristic
186Q16×C73—21no—
187Q16×C73—21no—
188D8×C73—21no—
189D8×C73—21no—
190C2×C8×C73—21no—
191C2×C8×C73—21no—
192C2×C8×C73C2×D411yescharacteristic
193C2×C8×C73C2411yescharacteristic
194C4⋊C4×C73—21no—
195C4⋊C4×C73—21no—
196C4⋊C4×C73—41no—
197C22⋊C4×C73—41no—
198C22⋊C4×C73—41no—
199C16×C73—21no—
200C16×C73C2×D412yescomplemented
201C23×C73—41no—
202C23×C73C2×D811yescharacteristic
203C22×D73—22no—
204C22×D73—22no—
205C22×D73—41no—
206C22×D73—82no—
207C22×D73—82no—
208C22×D73—82no—
209C22×D73SD3214yescomplemented
210C22×D73C2×D811yescharacteristic
211C22×D73C2×D811yescharacteristic
212Q8×C73—41no—
213Q8×C73—41no—
214D4×C73—41no—
215D4×C73—41no—
216D4×C73—82no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—41no—
220C2×C4×C73—41no—
221C2×C4×C73—41no—
222C2×C4×C73C2×D811yescharacteristic
223C2×C4×C73C22×D411yescharacteristic
224C22⋊D73—82no—
225C22⋊D73—82no—
226C22⋊D73—82no—
227C2×Dic73—22no—
228C2×Dic73—41no—
229C2×Dic73—41no—
230C2×Dic73—41no—
231C2×Dic73C2×D811yescharacteristic
232C2×Dic73C2×D811yescharacteristic
233D292—41no—
234D292—41no—
235D292—82no—
236C4×D73—22no—
237C4×D73—21no—
238C4×D73—21no—
239C4×D73—22no—
240C4×D73—82no—
241C4×D73—82no—
242C4×D73—82no—
243C4×D73#12112yes—
244C4×D73#13512yes—
245Dic146—41no—
246Dic146—41no—
247Dic146—82no—
248C8×C73—21no—
249C8×C73C22×D411yescharacteristic
250C8×C73C2×#4111yescharacteristic
251#10596831—21no—
252#10596831#12112yes—
253C22×C73—21no—
254C22×C73—21no—
255C22×C73—81no—
256C22×C73—81no—
257C22×C73—81no—
258C22×C73C2×SD3212yescomplemented
259C22×C73C22×D811yescharacteristic
260C2×D73—21no—
261C2×D73—21no—
262C2×D73—28no—
263C2×D73—81no—
264C2×D73—81no—
265C2×D73—81no—
266C2×D73—164no—
267C2×D73#46612yes—
268C2×D73C2×SD3212yes—
269C2×D73C2×SD3212yescomplemented
270C2×D73#50812yes—
271C4×C73—21no—
272C4×C73—81no—
273C4×C73—81no—
274C4×C73—81no—
275C4×C73C2×#12111yescharacteristic
276C4×C73C2×#13511yescharacteristic
277Dic73—21no—
278Dic73—81no—
279Dic73—81no—
280Dic73—81no—
281Dic73#46612yes—
282C2×#2217—731nomaximal, Sylow
283C2×C73—22no—
284C2×C73—161no—
285C2×C73C2×#46611yescharacteristic
286C2×C73C22×SD3211yescharacteristic
287C2×C73C2×#50811yescharacteristic
288D73—22no—
289D73—161no—
290D73#221714yesdirect product factor
291C22×SD32—731no—
292C23×C16—731no—
293C2×#467—731no—
294C2×#466—731no—
295#2217—731no—
296#2217—731no—
297#2217—731no—
298#2217—731no—
299#2217—731no—
300#2217—731no—
301#2217—731no—
302#2217D7311yescharacteristic, direct product factor
303C2×#367—731no—
304C2×#358—731no—
305C2×#357—731no—
306C73C2×#221711yescharacteristic, complemented, Sylow
307C22×Q16—731no—
308C22×D8—731no—
309C23×C8—731no—
310C2×C22⋊Q8—1461no—
311C2×#121—1461no—
312C2×SD32—731no—
313C2×SD32—731no—
314C2×SD32—731no—
315C2×SD32—1464no—
316C2×SD32C2×D7311yescharacteristic, complemented
317C22×C16—731no—
318C22×C16—731no—
319C22×C16—731no—
320C22×C16—731no—
321C22×C16—731no—
322C22×C16—1464no—
323C22×C16C2×D7311yescharacteristic, complemented
324#467—731no—
325#467—731no—
326#467—731no—
327#467—1461no—
328#467—1461no—
329#467C2×D7311yescharacteristic, complemented
330#466—731no—
331#466—731no—
332#466—731no—
333#466—1461no—
334#466—1461no—
335#466C2×D7311yescharacteristic
336C2×#107—731no—
337C2×#107—731no—
338C2×#103—1461no—
339C2×#102—1461no—
340#367—731no—
341#367—731no—
342#367—731no—
343#367C2×D7311yescharacteristic, complemented
344#358—731no—
345#358—731no—
346#358—731no—
347#358C2×D7311yescharacteristic, complemented
348#357—731no—
349#357—731no—
350#357—731no—
351#357C2×D7311yescharacteristic, complemented
352C22×Q8—1461no—
353C22×D4—2921no—
354C22×D4—1461no—
355C23×C4—731no—
356C2×Q16—731no—
357C2×Q16—1462no—
358C2×Q16—1462no—
359C2×Q16—1461no—
360C2×Q16C22×D7311yescharacteristic
361C2×D8—731no—
362C2×D8—1462no—
363C2×D8—1462no—
364C2×D8—1461no—
365C2×D8C22×D7311yescharacteristic
366C22×C8—731no—
367C22×C8—731no—
368C22×C8—731no—
369C22×C8—731no—
370C22×C8—731no—
371C22×C8—1462no—
372C22×C8—1462no—
373C22×C8C22×D7311yescharacteristic
374C22⋊Q8—21no—
375C22⋊Q8—2921no—
376C22⋊Q8—2921no—
377C22⋊Q8—1461no—
378C22⋊Q8—1461no—
379C22⋊Q8—1461no—
380#121—21no—
381#121—2921no—
382#121—2921no—
383#121—1461no—
384#121—1461no—
385#121—1461no—
386C2×C4⋊C4—2921no—
387C2×C4⋊C4—1461no—
388C2×C4⋊C4—1461no—
389C2×C22⋊C4—2921no—
390C2×C22⋊C4—2921no—
391SD32—22no—
392SD32—1462no—
393SD32—1462no—
394SD32—1462no—
395C2×C16—22no—
396C2×C16—731no—
397C2×C16—734no—
398C2×C16—731no—
399C2×C16—1462no—
400C2×C16—1462no—
401C2×C16—1462no—
402C2×C16—1462no—
403C2×C16C22×D7311yescharacteristic
404C2×C16C22×D7311yescharacteristic
405#107—731no—
406#107—731no—
407#107—1461no—
408#107—1461no—
409#107C22×D7311yescharacteristic
410#107C22×D7311yescharacteristic
411#103—21no—
412#103—1461no—
413#103—1461no—
414#103—1461no—
415#102—21no—
416#102—1461no—
417#102—1461no—
418#102—1461no—
419C24—2921no—
420C24—731no—
421C2×Q8—21no—
422C2×Q8—2921no—
423C2×Q8—2922no—
424C2×Q8—2922no—
425C2×Q8—1461no—
426C2×D4—21no—
427C2×D4—41no—
428C2×D4—2921no—
429C2×D4—2921no—
430C2×D4—2921no—
431C2×D4—2922no—
432C2×D4—2922no—
433C2×D4—2921no—
434C2×D4—5844no—
435C2×D4—1461no—
436C22×C4—2921no—
437C22×C4—2921no—
438C22×C4—2921no—
439C22×C4—731no—
440C22×C4—731no—
441C22×C4—731no—
442C22×C4—731no—
443C22×C4—731no—
444C22×C4—1462no—
445C22×C4—1462no—
446C22×C4D4×D7311yescharacteristic
447Q16—21no—
448Q16—21no—
449Q16—2922no—
450Q16—1461no—
451Q16—1461no—
452D8—21no—
453D8—21no—
454D8—2922no—
455D8—1461no—
456D8—1461no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—731no—
460C2×C8—732no—
461C2×C8—732no—
462C2×C8—731no—
463C2×C8—1462no—
464C2×C8—1462no—
465C2×C8—1462no—
466C2×C8—1461no—
467C2×C8—1461no—
468C2×C8D4×D7311yescharacteristic
469C2×C8C23×D7311yescharacteristic
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—41no—
473C4⋊C4—2921no—
474C4⋊C4—2921no—
475C4⋊C4—2921no—
476C4⋊C4—2921no—
477C4⋊C4—2921no—
478C4⋊C4—1461no—
479C4⋊C4—1461no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—2921no—
483C22⋊C4—2921no—
484C22⋊C4—2921no—
485C22⋊C4—2921no—
486C22⋊C4—2921no—
487C22⋊C4—2921no—
488C16—21no—
489C16—732no—
490C16—1461no—
491C16D4×D7312yescomplemented
492C23—41no—
493C23—2921no—
494C23—5842no—
495C23—5842no—
496C23—5842no—
497C23—731no—
498C23—734no—
499C23—731no—
500C23—1462no—
501C23—1462no—
502C23D8×D7311yescharacteristic
503Q8—41no—
504Q8—41no—
505Q8—2921no—
506Q8—2921no—
507Q8—5842no—
508D4—41no—
509D4—41no—
510D4—82no—
511D4—2921no—
512D4—2921no—
513D4—5842no—
514D4—5842no—
515D4—5842no—
516D4—5842no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—2921no—
523C2×C4—2921no—
524C2×C4—2921no—
525C2×C4—5842no—
526C2×C4—5842no—
527C2×C4—5842no—
528C2×C4—731no—
529C2×C4—732no—
530C2×C4—732no—
531C2×C4—731no—
532C2×C4—1462no—
533C2×C4—1462no—
534C2×C4—1462no—
535C2×C4—1461no—
536C2×C4—1461no—
537C2×C4D8×D7311yescharacteristic
538C2×C4C2×D4×D7311yescharacteristic
539C8—21no—
540C8—732no—
541C8—1461no—
542C8C2×D4×D7311yescharacteristic
543C8#41×D7311yescharacteristic
544C22—21no—
545C22—21no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22—5841no—
550C22—5841no—
551C22—5841no—
552C22—732no—
553C22—732no—
554C22—732no—
555C22—732no—
556C22—11684no—
557C22—1468no—
558C22—1461no—
559C22—1461no—
560C22SD32×D7312yescomplemented
561C22C2×D8×D7311yescharacteristic
562C4—21no—
563C4—81no—
564C4—81no—
565C4—81no—
566C4—5841no—
567C4—5841no—
568C4—5841no—
569C4—732no—
570C4—1461no—
571C4#121×D7311yescharacteristic
572C4#135×D7311yescharacteristic
573C2—22no—
574C2—161no—
575C2—734no—
576C2—11681no—
577C2—1462no—
578C2#466×D7311yescharacteristic
579C2C2×SD32×D7311yescharacteristic
580C2#508×D7311yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#2217C2
socleC22×C73C22×D8
prime radical#466×D73C2
hypercenter#2217D73
hypocenterC73C2×#2217
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2217

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×SD32×D73C2semidirect
2. C22×C16×D73C2semidirect
3. #467×D73C2semidirect
5. #22612030C2semidirect
6. #22612029C2semidirect
7. #22612028C2semidirect
8. #22612027C2semidirect
9. #22612026C2semidirect
10. #22612025C2semidirect
11. #22612024C2semidirect
12. #367×D73C2semidirect
13. #358×D73C2semidirect
14. #357×D73C2semidirect
15. C73×#2217C2semidirect
21. C2×#14556536C22semidirect
22. C2×#14556535C22semidirect
23. C2×#14556534C22semidirect
25. C22×#11924003C22semidirect
29. C2×C16×D73C22semidirect
30. C2×C16×D73C22semidirect
31. #17988829C22semidirect
32. #17988828C22semidirect
35. #17988825C22semidirect
45. #17988415C22semidirect
46. #17988414C22semidirect
47. #17988414C22semidirect
48. #17988392C22semidirect
49. #17988391C22semidirect
50. #17988390C22semidirect
51. #17988389C22semidirect
52. #17988388C22semidirect
53. #17988387C22semidirect
54. C2×SD32×C73C22semidirect
55. C22×C16×C73C22semidirect
56. #467×C73C22semidirect
58. #367×C73C22semidirect
59. #358×C73C22semidirect
60. #357×C73C22semidirect
104. C16×D73D4semidirect
181. #11924003C2×D4semidirect
200. C16×C73C2×D4semidirect
209. C22×D73SD32semidirect
258. C22×C73C2×SD32semidirect
269. C2×D73C2×SD32semidirect
290. D73#2217direct (N × H)
302. #2217D73direct (N × H)
306. C73C2×#2217semidirect
316. C2×SD32C2×D73semidirect
323. C22×C16C2×D73semidirect
329. #467C2×D73semidirect
343. #367C2×D73semidirect
347. #358C2×D73semidirect
351. #357C2×D73semidirect
491. C16D4×D73semidirect
560. C22SD32×D73semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×SD32×D732
C22×C16×D732
#467×D732
#466×D732
#226120302
#226120292
#226120282
#226120272
#226120262
#226120252
#226120242
#367×D732
#358×D732
#357×D732
C73×#22172
C2×#221773